Question: Determine how many solutions exist for the system of equations. ${-12x-2y = -14}$ ${6x+y = 7}$
Explanation: Convert both equations to slope-intercept form: ${-12x-2y = -14}$ $-12x{+12x} - 2y = -14{+12x}$ $-2y = -14+12x$ $y = 7-6x$ ${y = -6x+7}$ ${6x+y = 7}$ $6x{-6x} + y = 7{-6x}$ $y = 7-6x$ ${y = -6x+7}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -6x+7}$ ${y = -6x+7}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-12x-2y = -14}$ is also a solution of ${6x+y = 7}$, there are infinitely many solutions.